Seite:Hans Euler Über die Streuung von Licht an Licht nach der Diracschen Theorie 1936.pdf/37

	Hans Heinrich Euler: Über die Streuung von Licht an Licht nach der Diracschen Theorie

Gl. (8,10)

${\textrm {Reihen-}}$ ${\textrm {folge}}$

${\mbox{Zähler}}$

$1$

${\frac {({\mathfrak {pg}})}{{p_{0}}^{2}}}$

${\frac {|g|^{2}}{{p_{0}}^{2}}}$

${\frac {({\mathfrak {pg}})^{2}}{p_{0}^{4}}}$

${\frac {({\mathfrak {pg}})|g|^{2}}{{p_{0}}^{4}}}$

${\frac {({\mathfrak {pg}})^{3}}{{p_{0}}^{6}}}$

${\frac {|g|^{4}}{{p_{0}}^{4}}}$

${\frac {({\mathfrak {pg}})^{2}|g|^{2}}{{p_{0}}^{6}}}$

${\frac {({\mathfrak {pg}})^{4}}{{p_{0}}^{8}}}$

$1$

${\frac {2{p_{y}}^{2}}{{p_{0}}^{2}}}$

${\frac {8{p_{y}}^{2}{p_{z}}^{2}}{{p_{0}}^{4}}}$

$1$

${\frac {2{p_{y}}^{2}}{{p_{0}}^{2}}}$

${\frac {8{p_{y}}^{2}{p_{z}}^{2}}{{p_{0}}^{4}}}$

$1$

${\frac {2{p_{y}}^{2}}{{p_{0}}^{2}}}$

${\frac {8{p_{y}}^{2}{p_{z}}^{2}}{{p_{0}}^{4}}}$

$1$

${\frac {2{p_{y}}^{2}}{{p_{0}}^{2}}}$

${\frac {8{p_{y}}^{2}{p_{z}}^{2}}{{p_{0}}^{4}}}$

$1$

${\frac {2{p_{y}}^{2}}{{p_{0}}^{2}}}$

${\frac {8{p_{y}}^{2}{p_{z}}^{2}}{{p_{0}}^{4}}}$

$1$

${\frac {2{p_{y}}^{2}}{{p_{0}}^{2}}}$

${\frac {8{p_{y}}^{2}{p_{z}}^{2}}{{p_{0}}^{4}}}$

$1$

${\frac {2{p_{y}}^{2}}{{p_{0}}^{2}}}$

${\frac {8{p_{y}}^{2}{p_{z}}^{2}}{{p_{0}}^{4}}}$

$1$

${\frac {2{p_{y}}^{2}}{{p_{0}}^{2}}}$

${\frac {8{p_{y}}^{2}{p_{z}}^{2}}{{p_{0}}^{4}}}$

$1$

${\frac {2{p_{y}}^{2}}{{p_{0}}^{2}}}$

${\frac {8{p_{y}}^{2}{p_{z}}^{2}}{{p_{0}}^{4}}}$

$g^{1}g^{2}g^{3}g^{4}$

${\begin{aligned}z_{1}=\\z_{2}=z_{3}=\\z_{4}=\\z_{5}=z_{6}=\end{aligned}}$

${\begin{aligned}0\\0\\0\\-8\end{aligned}}$

${\begin{aligned}0\\0\\4\\8\end{aligned}}$

${\begin{aligned}-1\\-1\\-1\\-1\end{aligned}}$

${\begin{aligned}0\\0\\0\\0\end{aligned}}$

${\begin{aligned}4\\4\\-4\\-8\end{aligned}}$

${\begin{aligned}2\\2\\2\\2\end{aligned}}$

${\begin{aligned}2\\2\\2\\4\end{aligned}}$

${\begin{aligned}0\\0\\-4\\-6\end{aligned}}$

${\begin{aligned}1\\1\\1\\1\end{aligned}}$

${\begin{aligned}-2\\-2\\-2\\-4\end{aligned}}$

${\begin{aligned}-8\\-8\\8\\14\end{aligned}}$

${\begin{aligned}-4\\-4\\-4\\-4\end{aligned}}$

${\begin{aligned}-4\\-4\\-4\\-8\end{aligned}}$

${\begin{aligned}-4\\-4\\12\\18\end{aligned}}$

${\begin{aligned}-4\\-4\\-4\\-4\end{aligned}}$

${\begin{aligned}4\\4\\4\\+8\end{aligned}}$

${\begin{aligned}16\\16\\-16\\-26\end{aligned}}$

${\begin{aligned}8\\8\\8\\8\end{aligned}}$

${\begin{aligned}-2\\-2\\-2\\-{\tfrac {7}{2}}\end{aligned}}$

${\begin{aligned}0\\0\\4\\{\tfrac {11}{2}}\end{aligned}}$

${\begin{aligned}-1\\-1\\-1\\-1\end{aligned}}$

${\begin{aligned}10\\10\\10\\19\end{aligned}}$

${\begin{aligned}16\\16\\-32\\-47\end{aligned}}$

${\begin{aligned}12\\12\\12\\12\end{aligned}}$

${\begin{aligned}-8\\-8\\-8\\-{\tfrac {31}{2}}\end{aligned}}$

${\begin{aligned}-32\\-32\\+32\\{\tfrac {99}{2}}\end{aligned}}$

${\begin{aligned}-16\\-16\\-16\\-16\end{aligned}}$

$g^{1}g^{2}g^{4}g^{3}$

${\begin{aligned}z_{1}=\\z_{2}=z_{3}=\\z_{4}=\\z_{5}=z_{6}=\end{aligned}}$

${\begin{aligned}0\\0\\0\\-8\end{aligned}}$

${\begin{aligned}0\\0\\4\\8\end{aligned}}$

${\begin{aligned}-1\\-1\\-1\\-1\end{aligned}}$

${\begin{aligned}0\\0\\0\\0\end{aligned}}$

${\begin{aligned}0\\0\\4\\0\end{aligned}}$

${\begin{aligned}0\\0\\0\\0\end{aligned}}$

${\begin{aligned}-2\\-2\\2\\4\end{aligned}}$

${\begin{aligned}0\\0\\-4\\-6\end{aligned}}$

${\begin{aligned}1\\1\\1\\1\end{aligned}}$

${\begin{aligned}2\\2\\-2\\-4\end{aligned}}$

${\begin{aligned}0\\0\\8\\14\end{aligned}}$

${\begin{aligned}-2\\-2\\-2\\-2\end{aligned}}$

${\begin{aligned}0\\0\\4\\0\end{aligned}}$

${\begin{aligned}0\\0\\-6\\0\end{aligned}}$

${\begin{aligned}0\\0\\0\\0\end{aligned}}$

${\begin{aligned}0\\0\\-4\\0\end{aligned}}$

${\begin{aligned}0\\0\\10\\0\end{aligned}}$

${\begin{aligned}0\\0\\0\\0\end{aligned}}$

${\begin{aligned}2\\2\\-2\\-{\tfrac {7}{2}}\end{aligned}}$

${\begin{aligned}0\\0\\4\\{\tfrac {11}{2}}\end{aligned}}$

${\begin{aligned}-1\\-1\\-1\\-1\end{aligned}}$

${\begin{aligned}-6\\-6\\10\\19\end{aligned}}$

${\begin{aligned}0\\0\\-24\\-39\end{aligned}}$

${\begin{aligned}6\\6\\6\\6\end{aligned}}$

${\begin{aligned}4\\4\\-8\\-{\tfrac {31}{2}}\end{aligned}}$

${\begin{aligned}0\\0\\24\\{\tfrac {83}{2}}\end{aligned}}$

${\begin{aligned}-6\\-6\\-6\\-6\end{aligned}}$

$g^{1}g^{3}g^{4}g^{2}$

${\begin{aligned}z_{1}=\\z_{2}=z_{3}=\\z_{4}=\\z_{5}=z_{6}=\end{aligned}}$

${\begin{aligned}0\\0\\0\\-8\end{aligned}}$

${\begin{aligned}4\\0\\0\\8\end{aligned}}$

${\begin{aligned}-1\\-1\\-1\\-1\end{aligned}}$

${\begin{aligned}0\\0\\0\\0\end{aligned}}$

${\begin{aligned}-12\\0\\0\\-16\end{aligned}}$

${\begin{aligned}4\\4\\4\\4\end{aligned}}$

${\begin{aligned}2\\-2\\-2\\4\end{aligned}}$

${\begin{aligned}-12\\0\\0\\-14\end{aligned}}$

${\begin{aligned}3\\3\\3\\3\end{aligned}}$

${\begin{aligned}-2\\2\\2\\-4\end{aligned}}$

${\begin{aligned}40\\0\\0\\46\end{aligned}}$

${\begin{aligned}-14\\-14\\-14\\-14\end{aligned}}$

${\begin{aligned}-12\\8\\8\\-16\end{aligned}}$

${\begin{aligned}78\\0\\0\\84\end{aligned}}$

${\begin{aligned}-22\\-22\\-22\\-22\end{aligned}}$

${\begin{aligned}12\\-8\\-8\\16\end{aligned}}$

${\begin{aligned}-138\\0\\0\\-148\end{aligned}}$

${\begin{aligned}48\\48\\48\\48\end{aligned}}$

${\begin{aligned}-10\\6\\6\\-{\tfrac {23}{2}}\end{aligned}}$

${\begin{aligned}38\\0\\0\\{\tfrac {79}{2}}\end{aligned}}$

${\begin{aligned}-9\\-9\\-9\\-9\end{aligned}}$

${\begin{aligned}66\\-34\\-34\\75\end{aligned}}$

${\begin{aligned}-402\\0\\0\\-417\end{aligned}}$

${\begin{aligned}118\\118\\118\\118\end{aligned}}$

${\begin{aligned}-56\\28\\28\\-{\tfrac {127}{2}}\end{aligned}}$

${\begin{aligned}488\\0\\0\\{\tfrac {1011}{2}}\end{aligned}}$

${\begin{aligned}-166\\-166\\-166\\-166\end{aligned}}$

$g^{1}g^{4}g^{3}g^{2}$

${\begin{aligned}z_{1}=\\z_{2}=z_{3}=\\z_{4}=\\z_{5}=z_{6}=\end{aligned}}$

${\begin{aligned}0\\0\\0\\-8\end{aligned}}$

${\begin{aligned}4\\0\\0\\8\end{aligned}}$

${\begin{aligned}-1\\-1\\-1\\-1\end{aligned}}$

${\begin{aligned}0\\0\\0\\0\end{aligned}}$

${\begin{aligned}-4\\-4\\-4\\-8\end{aligned}}$

${\begin{aligned}2\\2\\2\\2\end{aligned}}$

${\begin{aligned}2\\2\\2\\4\end{aligned}}$

${\begin{aligned}-4\\-4\\-4\\-6\end{aligned}}$

${\begin{aligned}1\\1\\1\\1\end{aligned}}$

${\begin{aligned}-2\\-2\\-2\\-4\end{aligned}}$

${\begin{aligned}8\\8\\8\\14\end{aligned}}$

${\begin{aligned}-4\\-4\\-4\\-4\end{aligned}}$

${\begin{aligned}-4\\-4\\-4\\-8\end{aligned}}$

${\begin{aligned}12\\12\\12\\18\end{aligned}}$

${\begin{aligned}-4\\-4\\-4\\-4\end{aligned}}$

${\begin{aligned}4\\4\\4\\8\end{aligned}}$

${\begin{aligned}-16\\-16\\-16\\-26\end{aligned}}$

${\begin{aligned}8\\8\\8\\8\end{aligned}}$

${\begin{aligned}-2\\-2\\-2\\-{\tfrac {7}{2}}\end{aligned}}$

${\begin{aligned}4\\4\\4\\{\tfrac {11}{2}}\end{aligned}}$

${\begin{aligned}-1\\-1\\-1\\-1\end{aligned}}$

${\begin{aligned}10\\10\\10\\19\end{aligned}}$

${\begin{aligned}-32\\-32\\-32\\-47\end{aligned}}$

${\begin{aligned}12\\12\\12\\12\end{aligned}}$

${\begin{aligned}-8\\-8\\-8\\-{\tfrac {31}{2}}\end{aligned}}$

${\begin{aligned}32\\32\\32\\{\tfrac {99}{2}}\end{aligned}}$

${\begin{aligned}-16\\-16\\-16\\-16\end{aligned}}$

$g^{1}g^{4}g^{2}g^{3}$

${\begin{aligned}z_{1}=\\z_{2}=z_{3}=\\z_{4}=\\z_{5}=z_{6}=\end{aligned}}$

${\begin{aligned}0\\0\\0\\8\end{aligned}}$

${\begin{aligned}0\\4\\0\\-8\end{aligned}}$

${\begin{aligned}-1\\-1\\-1\\-1\end{aligned}}$

${\begin{aligned}0\\0\\0\\0\end{aligned}}$

${\begin{aligned}0\\0\\-4\\0\end{aligned}}$

${\begin{aligned}0\\0\\0\\0\end{aligned}}$

${\begin{aligned}2\\2\\-2\\-4\end{aligned}}$

${\begin{aligned}-4\\-4\\0\\2\end{aligned}}$

${\begin{aligned}1\\1\\1\\1\end{aligned}}$

${\begin{aligned}-2\\-2\\2\\4\end{aligned}}$

${\begin{aligned}4\\4\\-4\\-10\end{aligned}}$

${\begin{aligned}-2\\-2\\-2\\-2\end{aligned}}$

${\begin{aligned}0\\0\\-4\\0\end{aligned}}$

${\begin{aligned}0\\0\\6\\0\end{aligned}}$

${\begin{aligned}0\\0\\0\\0\end{aligned}}$

${\begin{aligned}0\\0\\4\\0\end{aligned}}$

${\begin{aligned}0\\0\\-10\\0\end{aligned}}$

${\begin{aligned}0\\0\\0\\0\end{aligned}}$

${\begin{aligned}-2\\-2\\2\\{\tfrac {7}{2}}\end{aligned}}$

${\begin{aligned}4\\4\\0\\-{\tfrac {3}{2}}\end{aligned}}$

${\begin{aligned}-1\\-1\\-1\\-1\end{aligned}}$

${\begin{aligned}6\\6\\-10\\-19\end{aligned}}$

${\begin{aligned}-16\\-16\\8\\23\end{aligned}}$

${\begin{aligned}6\\6\\6\\6\end{aligned}}$

${\begin{aligned}-4\\-4\\8\\{\tfrac {31}{2}}\end{aligned}}$

${\begin{aligned}12\\12\\-12\\-{\tfrac {59}{2}}\end{aligned}}$

${\begin{aligned}-6\\-6\\-6\\-6\end{aligned}}$

$g^{1}g^{3}g^{2}g^{4}$

${\begin{aligned}z_{1}=\\z_{2}=z_{3}=\\z_{4}=\\z_{5}=z_{6}=\end{aligned}}$

${\begin{aligned}0\\0\\0\\8\end{aligned}}$

${\begin{aligned}0\\4\\0\\-8\end{aligned}}$

${\begin{aligned}-1\\-1\\-1\\-1\end{aligned}}$

${\begin{aligned}0\\0\\0\\0\end{aligned}}$

${\begin{aligned}4\\-8\\0\\16\end{aligned}}$

${\begin{aligned}4\\4\\4\\4\end{aligned}}$

${\begin{aligned}-2\\2\\2\\-4\end{aligned}}$

${\begin{aligned}4\\-8\\-4\\10\end{aligned}}$

${\begin{aligned}3\\3\\3\\3\end{aligned}}$

${\begin{aligned}2\\-2\\-2\\4\end{aligned}}$

${\begin{aligned}-20\\20\\4\\-42\end{aligned}}$

${\begin{aligned}-14\\-14\\-14\\-14\end{aligned}}$

${\begin{aligned}12\\-8\\-8\\16\end{aligned}}$

${\begin{aligned}-38\\40\\24\\-60\end{aligned}}$

${\begin{aligned}-22\\-22\\-22\\-22\end{aligned}}$

${\begin{aligned}-12\\8\\8\\-16\end{aligned}}$

${\begin{aligned}82\\-56\\-24\\124\end{aligned}}$

${\begin{aligned}48\\48\\48\\48\end{aligned}}$

${\begin{aligned}10\\-6\\-6\\{\tfrac {23}{2}}\end{aligned}}$

${\begin{aligned}-18\\20\\16\\-{\tfrac {47}{2}}\end{aligned}}$

${\begin{aligned}-9\\-9\\-9\\-9\end{aligned}}$

${\begin{aligned}-66\\34\\34\\-75\end{aligned}}$

${\begin{aligned}230\\-172\\-124\\293\end{aligned}}$

${\begin{aligned}118\\118\\118\\118\end{aligned}}$

${\begin{aligned}56\\-28\\-28\\{\tfrac {127}{2}}\end{aligned}}$

${\begin{aligned}-316\\172\\108\\-{\tfrac {795}{2}}\end{aligned}}$

${\begin{aligned}-166\\-166\\-166\\-166\end{aligned}}$

Für die Reihenfolge: $g^{1}g^{4}g^{2}g^{3}$ und $g^{1}g^{3}g^{2}g^{4}$

${\begin{aligned}Z_{\mu }&=-S_{1}+2{p_{y}}^{2}\left[-{\frac {{\lambda _{\mu }}^{3}}{{p_{0}}^{1}{p_{0}}^{2}}}+{\frac {{\lambda _{\mu }}^{2}}{{p_{0}}^{3}{p_{0}}^{4}}}+{\frac {{\lambda _{\mu }}^{2}{\lambda _{\mu }}^{3}}{{p_{0}}^{1}{p_{0}}^{4}}}-{\frac {1}{{p_{0}}^{2}{p_{0}}^{3}}}+{\frac {2{\lambda _{\mu }}^{2}}{{p_{0}}^{1}{p_{0}}^{3}}}-{\frac {2{\lambda _{\mu }}^{3}}{{p_{0}}^{2}{p_{0}}^{4}}}\right]\\&-{\frac {1}{{p_{0}}^{1}{p_{0}}^{2}{p_{0}}^{3}{p_{0}}^{4}}}\left[-S_{2}+2{p_{y}}^{2}\left[\left({\mathfrak {p}}^{1}{\mathfrak {p}}^{2}\right)+\left({\mathfrak {p}}^{3}{\mathfrak {p}}^{4}\right)+\left({\mathfrak {p}}^{2}{\mathfrak {p}}^{3}\right)+\left({\mathfrak {p}}^{1}{\mathfrak {p}}^{4}\right)-2\left({\mathfrak {p}}^{1}{\mathfrak {p}}^{3}\right)-2\left({\mathfrak {p}}^{1}{\mathfrak {p}}^{4}\right)\right]\right]\end{aligned}}$

Darin bedeuten $S_{1}$, $S_{2}$ die Abkürzung:

${\begin{aligned}S_{1}&={\lambda _{\mu }}^{2}{\lambda _{\mu }}^{3}-{\lambda _{\mu }}^{3}{\frac {\left({\mathfrak {p}}^{1}{\mathfrak {p}}^{2}\right)+(mc)^{2}}{{p_{0}}^{1}{p_{0}}^{2}}}+{\lambda _{\mu }}^{2}{\frac {\left({\mathfrak {p}}^{3}{\mathfrak {p}}^{4}\right)+(mc)^{2}}{{p_{0}}^{3}{p_{0}}^{4}}}-{\lambda _{\mu }}^{3}{\frac {\left({\mathfrak {p}}^{2}{\mathfrak {p}}^{4}\right)+(mc)^{2}}{{p_{0}}^{2}{p_{0}}^{4}}}\\&+{\lambda _{\mu }}^{2}{\lambda _{\mu }}^{3}{\frac {\left({\mathfrak {p}}^{1}{\mathfrak {p}}^{4}\right)+(mc)^{2}}{{p_{0}}^{1}{p_{0}}^{4}}}-1\cdot {\frac {\left({\mathfrak {p}}^{2}{\mathfrak {p}}^{3}\right)+(mc)^{2}}{{p_{0}}^{2}{p_{0}}^{3}}}+{\lambda _{\mu }}^{2}{\frac {\left({\mathfrak {p}}^{1}{\mathfrak {p}}^{3}\right)+(mc)^{2}}{{p_{0}}^{1}{p_{0}}^{3}}}\end{aligned}}$,

$S_{2}=\left[{\begin{aligned}&\left[\left({\mathfrak {p}}^{1}{\mathfrak {p}}^{2}\right)+(mc)^{2}\right]\left[\left({\mathfrak {p}}^{3}{\mathfrak {p}}^{4}\right)+(mc)^{2}\right]+\left[\left({\mathfrak {p}}^{2}{\mathfrak {p}}^{3}\right)+(mc)^{2}\right]\left[\left({\mathfrak {p}}^{1}{\mathfrak {p}}^{4}\right)+(mc)^{2}\right]\\-&\left[\left({\mathfrak {p}}^{1}{\mathfrak {p}}^{3}\right)+(mc)^{2}\right]\left[\left({\mathfrak {p}}^{2}{\mathfrak {p}}^{4}\right)+(mc)^{2}\right]\end{aligned}}\right]$.

Diese Ausdrücke werden bei weiterer Spezialisierung auf parallele Impulse, wie man durch Einsetzen von (8‚8‚ $\delta$) und Entwicklung nach $g/mc$ bis zur 4. Ordnung findet:

Vgl. Gl. (8,10).

Um die Zähler (5,15) durch die Nenner (5,14) zu dividieren, über die 6 Übergangswege $\mu$ und die 24 Reihenfolgen (8,8, $\alpha$) zu summieren, muß man die Spalten der Tab. (8,10) mit den entsprechenden Spalten der Tab. (8,9) und mit dem Faktor 4 multiplizieren. Man erhält dann im Glied 4. Ordnung in $g/mc$ nach Mittelung über die Positronenrichtung ${\mathfrak {p}}$: [vgl. § 9,7]

Empfohlene Zitierweise:
Hans Heinrich Euler: Über die Streuung von Licht an Licht nach der Diracschen Theorie. , 1936, Seite 432. Digitale Volltext-Ausgabe bei Wikisource, URL: https://de.wikisource.org/w/index.php?title=Seite:Hans_Euler_%C3%9Cber_die_Streuung_von_Licht_an_Licht_nach_der_Diracschen_Theorie_1936.pdf/37&oldid=- (Version vom 1.8.2018)